Hyperbolic Graphs of Small Complexity

In this paper we enumerate and classify the "simplest" pairs (M, G) where M is a closed orientable 3-manifold and G is a trivalent graph embedded in M. To enumerate the pairs we use a variation of Matveev's definition of complexity for 3-manifolds, and we consider only (0, 1, 2)-irreducible pairs, namely pairs (M, G) such that any 2-sphere in M intersecting G transversely in at most 2 points bounds a ball in M either disjoint from G or intersecting G in an unknotted arc. To classify the pairs our main tools are geometric invariants defined using hyperbolic geometry. In most cases, the graph complement admits a unique hyperbolic structure with parabolic meridians; this structure was computed and studied using Heard's program Orb and Goodman's program Snap. We determine all (0, 1, 2)-irreducible pairs up to complexity 5, allowing disconnected graphs but forbidding components without vertices in complexity 5. The result is a list of 129 pairs, of which 123 are hyp...